Nonadiabatic stage of damping of solitons and the intermediate asymptotics

Decay of Kadomtsev–Petviashvili lumps in dissipative media

“…1 by line 1, where  =  t. However, although the numerical data, which is shown by dots, follows the predicted decay rate, there is a noticeable shift of this data from the theoretical line. This can be explained by the influence of the initial non-adiabatic adjustment process, when a lump generates a small near-field perturbation (a "tail") and then decays adiabatically together with this tail (a 6 similar phenomenon has been studied in one-dimensional case for the KdV equations with small dissipative terms [14]). The insert in Fig.…”

Perturbed Solitons

“…In general, these additions make the equation non-integrable but, as already mentioned, the integrability is not a crucial factor for the direct perturbation theory. Examples of slowly varying solitons in different physical realizations were considered in many publications beginning from the early papers [8,9,16,31].…”

“…A more detailed analysis [7] shows that at T > 0 the radiation field in the vicinity of the soliton center has the form of the Airy function which broadens and oscillates behind the soliton. In [8] it was shown that in the case or viscous losses when in (6.14), q = m = 0, g ≠ 0, the formation of a "shelf" behind a soliton eventually transforms it to a triangle-shaped impulse attenuating according to the asymptotics of Burgers' equation which will be addressed in Chapter 8. In the same work the main stages of the process described above were experimentally observed in an electric line.…”

Decay of Benjamin–Ono solitons under the influence of dissipation

“…The convention (6) for (−ik) m ensures that F is real valued. It can then be presented in the alternative form as…”